LMFDB - Embedded newform 175.2.a.f.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [175,2,Mod(1,175)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("175.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(175, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 175 = 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	175.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,1,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$1.39738203537$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	175.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.56155 q^{2} +2.56155 q^{3} +0.438447 q^{4} -4.00000 q^{6} +1.00000 q^{7} +2.43845 q^{8} +3.56155 q^{9} +2.56155 q^{11} +1.12311 q^{12} -4.56155 q^{13} -1.56155 q^{14} -4.68466 q^{16} +4.56155 q^{17} -5.56155 q^{18} +1.12311 q^{19} +2.56155 q^{21} -4.00000 q^{22} +5.12311 q^{23} +6.24621 q^{24} +7.12311 q^{26} +1.43845 q^{27} +0.438447 q^{28} -5.68466 q^{29} +2.43845 q^{32} +6.56155 q^{33} -7.12311 q^{34} +1.56155 q^{36} -6.00000 q^{37} -1.75379 q^{38} -11.6847 q^{39} -3.12311 q^{41} -4.00000 q^{42} -9.12311 q^{43} +1.12311 q^{44} -8.00000 q^{46} -3.68466 q^{47} -12.0000 q^{48} +1.00000 q^{49} +11.6847 q^{51} -2.00000 q^{52} -3.12311 q^{53} -2.24621 q^{54} +2.43845 q^{56} +2.87689 q^{57} +8.87689 q^{58} -4.00000 q^{59} -9.36932 q^{61} +3.56155 q^{63} +5.56155 q^{64} -10.2462 q^{66} +6.24621 q^{67} +2.00000 q^{68} +13.1231 q^{69} +8.00000 q^{71} +8.68466 q^{72} -4.24621 q^{73} +9.36932 q^{74} +0.492423 q^{76} +2.56155 q^{77} +18.2462 q^{78} -6.56155 q^{79} -7.00000 q^{81} +4.87689 q^{82} -4.00000 q^{83} +1.12311 q^{84} +14.2462 q^{86} -14.5616 q^{87} +6.24621 q^{88} +7.12311 q^{89} -4.56155 q^{91} +2.24621 q^{92} +5.75379 q^{94} +6.24621 q^{96} +14.8078 q^{97} -1.56155 q^{98} +9.12311 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + q^{3} + 5 q^{4} - 8 q^{6} + 2 q^{7} + 9 q^{8} + 3 q^{9} + q^{11} - 6 q^{12} - 5 q^{13} + q^{14} + 3 q^{16} + 5 q^{17} - 7 q^{18} - 6 q^{19} + q^{21} - 8 q^{22} + 2 q^{23} - 4 q^{24} + 6 q^{26}+ \cdots + 10 q^{99}+O(q^{100}) $ 2 * q + q^2 + q^3 + 5 * q^4 - 8 * q^6 + 2 * q^7 + 9 * q^8 + 3 * q^9 + q^11 - 6 * q^12 - 5 * q^13 + q^14 + 3 * q^16 + 5 * q^17 - 7 * q^18 - 6 * q^19 + q^21 - 8 * q^22 + 2 * q^23 - 4 * q^24 + 6 * q^26 + 7 * q^27 + 5 * q^28 + q^29 + 9 * q^32 + 9 * q^33 - 6 * q^34 - q^36 - 12 * q^37 - 20 * q^38 - 11 * q^39 + 2 * q^41 - 8 * q^42 - 10 * q^43 - 6 * q^44 - 16 * q^46 + 5 * q^47 - 24 * q^48 + 2 * q^49 + 11 * q^51 - 4 * q^52 + 2 * q^53 + 12 * q^54 + 9 * q^56 + 14 * q^57 + 26 * q^58 - 8 * q^59 + 6 * q^61 + 3 * q^63 + 7 * q^64 - 4 * q^66 - 4 * q^67 + 4 * q^68 + 18 * q^69 + 16 * q^71 + 5 * q^72 + 8 * q^73 - 6 * q^74 - 32 * q^76 + q^77 + 20 * q^78 - 9 * q^79 - 14 * q^81 + 18 * q^82 - 8 * q^83 - 6 * q^84 + 12 * q^86 - 25 * q^87 - 4 * q^88 + 6 * q^89 - 5 * q^91 - 12 * q^92 + 28 * q^94 - 4 * q^96 + 9 * q^97 + q^98 + 10 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.56155	−1.10418	−0.552092	−	0.833783i	$-0.686170\pi$
$2$	−1.56155	−1.10418	−0.552092	+	0.833783i	$0.686170\pi$
$3$	2.56155	1.47891	0.739457	−	0.673204i	$-0.235083\pi$
$3$	2.56155	1.47891	0.739457	+	0.673204i	$0.235083\pi$
$4$	0.438447	0.219224
$5$	0	0
$5$	0	0
$6$	−4.00000	−1.63299
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.43845	0.862121
$9$	3.56155	1.18718
$10$	0	0
$11$	2.56155	0.772337	0.386169	−	0.922428i	$-0.373798\pi$
$11$	2.56155	0.772337	0.386169	+	0.922428i	$0.373798\pi$
$12$	1.12311	0.324213
$13$	−4.56155	−1.26515	−0.632574	−	0.774500i	$-0.718001\pi$
$13$	−4.56155	−1.26515	−0.632574	+	0.774500i	$0.718001\pi$
$14$	−1.56155	−0.417343
$15$	0	0
$16$	−4.68466	−1.17116
$17$	4.56155	1.10634	0.553170	−	0.833069i	$-0.313418\pi$
$17$	4.56155	1.10634	0.553170	+	0.833069i	$0.313418\pi$
$18$	−5.56155	−1.31087
$19$	1.12311	0.257658	0.128829	−	0.991667i	$-0.458878\pi$
$19$	1.12311	0.257658	0.128829	+	0.991667i	$0.458878\pi$
$20$	0	0
$21$	2.56155	0.558977
$22$	−4.00000	−0.852803
$23$	5.12311	1.06824	0.534121	−	0.845408i	$-0.320643\pi$
$23$	5.12311	1.06824	0.534121	+	0.845408i	$0.320643\pi$
$24$	6.24621	1.27500
$25$	0	0
$26$	7.12311	1.39696
$27$	1.43845	0.276829
$28$	0.438447	0.0828587
$29$	−5.68466	−1.05561	−0.527807	−	0.849364i	$-0.676986\pi$
$29$	−5.68466	−1.05561	−0.527807	+	0.849364i	$0.676986\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	2.43845	0.431061
$33$	6.56155	1.14222
$34$	−7.12311	−1.22160
$35$	0	0
$36$	1.56155	0.260259
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	−1.75379	−0.284502
$39$	−11.6847	−1.87104
$40$	0	0
$41$	−3.12311	−0.487747	−0.243874	−	0.969807i	$-0.578418\pi$
$41$	−3.12311	−0.487747	−0.243874	+	0.969807i	$0.578418\pi$
$42$	−4.00000	−0.617213
$43$	−9.12311	−1.39126	−0.695630	−	0.718400i	$-0.744875\pi$
$43$	−9.12311	−1.39126	−0.695630	+	0.718400i	$0.744875\pi$
$44$	1.12311	0.169315
$45$	0	0
$46$	−8.00000	−1.17954
$47$	−3.68466	−0.537463	−0.268731	−	0.963215i	$-0.586604\pi$
$47$	−3.68466	−0.537463	−0.268731	+	0.963215i	$0.586604\pi$
$48$	−12.0000	−1.73205
$49$	1.00000	0.142857
$50$	0	0
$51$	11.6847	1.63618
$52$	−2.00000	−0.277350
$53$	−3.12311	−0.428992	−0.214496	−	0.976725i	$-0.568811\pi$
$53$	−3.12311	−0.428992	−0.214496	+	0.976725i	$0.568811\pi$
$54$	−2.24621	−0.305671
$55$	0	0
$56$	2.43845	0.325851
$57$	2.87689	0.381054
$58$	8.87689	1.16559
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	−9.36932	−1.19962	−0.599809	−	0.800143i	$-0.704757\pi$
$61$	−9.36932	−1.19962	−0.599809	+	0.800143i	$0.704757\pi$
$62$	0	0
$63$	3.56155	0.448713
$64$	5.56155	0.695194
$65$	0	0
$66$	−10.2462	−1.26122
$67$	6.24621	0.763096	0.381548	−	0.924349i	$-0.375391\pi$
$67$	6.24621	0.763096	0.381548	+	0.924349i	$0.375391\pi$
$68$	2.00000	0.242536
$69$	13.1231	1.57984
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	8.68466	1.02350
$73$	−4.24621	−0.496981	−0.248491	−	0.968634i	$-0.579935\pi$
$73$	−4.24621	−0.496981	−0.248491	+	0.968634i	$0.579935\pi$
$74$	9.36932	1.08916
$75$	0	0
$76$	0.492423	0.0564847
$77$	2.56155	0.291916
$78$	18.2462	2.06598
$79$	−6.56155	−0.738232	−0.369116	−	0.929383i	$-0.620340\pi$
$79$	−6.56155	−0.738232	−0.369116	+	0.929383i	$0.620340\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	4.87689	0.538563
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	1.12311	0.122541
$85$	0	0
$86$	14.2462	1.53621
$87$	−14.5616	−1.56116
$88$	6.24621	0.665848
$89$	7.12311	0.755048	0.377524	−	0.926000i	$-0.376776\pi$
$89$	7.12311	0.755048	0.377524	+	0.926000i	$0.376776\pi$
$90$	0	0
$91$	−4.56155	−0.478181
$92$	2.24621	0.234184
$93$	0	0
$94$	5.75379	0.593458
$95$	0	0
$96$	6.24621	0.637501
$97$	14.8078	1.50350	0.751750	−	0.659448i	$-0.229210\pi$
$97$	14.8078	1.50350	0.751750	+	0.659448i	$0.229210\pi$
$98$	−1.56155	−0.157741
$99$	9.12311	0.916907
$100$	0	0
$101$	0.246211	0.0244989	0.0122495	−	0.999925i	$-0.496101\pi$
$101$	0.246211	0.0244989	0.0122495	+	0.999925i	$0.496101\pi$
$102$	−18.2462	−1.80664
$103$	−1.43845	−0.141734	−0.0708672	−	0.997486i	$-0.522577\pi$
$103$	−1.43845	−0.141734	−0.0708672	+	0.997486i	$0.522577\pi$
$104$	−11.1231	−1.09071
$105$	0	0
$106$	4.87689	0.473686
$107$	11.3693	1.09911	0.549557	−	0.835456i	$-0.314797\pi$
$107$	11.3693	1.09911	0.549557	+	0.835456i	$0.314797\pi$
$108$	0.630683	0.0606875
$109$	17.6847	1.69388	0.846942	−	0.531686i	$-0.178441\pi$
$109$	17.6847	1.69388	0.846942	+	0.531686i	$0.178441\pi$
$110$	0	0
$111$	−15.3693	−1.45879
$112$	−4.68466	−0.442659
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	−4.49242	−0.420754
$115$	0	0
$116$	−2.49242	−0.231416
$117$	−16.2462	−1.50196
$118$	6.24621	0.575010
$119$	4.56155	0.418157
$120$	0	0
$121$	−4.43845	−0.403495
$122$	14.6307	1.32460
$123$	−8.00000	−0.721336
$124$	0	0
$125$	0	0
$126$	−5.56155	−0.495463
$127$	−10.2462	−0.909204	−0.454602	−	0.890695i	$-0.650219\pi$
$127$	−10.2462	−0.909204	−0.454602	+	0.890695i	$0.650219\pi$
$128$	−13.5616	−1.19868
$129$	−23.3693	−2.05755
$130$	0	0
$131$	−9.12311	−0.797089	−0.398545	−	0.917149i	$-0.630485\pi$
$131$	−9.12311	−0.797089	−0.398545	+	0.917149i	$0.630485\pi$
$132$	2.87689	0.250402
$133$	1.12311	0.0973856
$134$	−9.75379	−0.842599
$135$	0	0
$136$	11.1231	0.953798
$137$	8.87689	0.758404	0.379202	−	0.925314i	$-0.376199\pi$
$137$	8.87689	0.758404	0.379202	+	0.925314i	$0.376199\pi$
$138$	−20.4924	−1.74443
$139$	−6.87689	−0.583291	−0.291645	−	0.956527i	$-0.594203\pi$
$139$	−6.87689	−0.583291	−0.291645	+	0.956527i	$0.594203\pi$
$140$	0	0
$141$	−9.43845	−0.794861
$142$	−12.4924	−1.04834
$143$	−11.6847	−0.977120
$144$	−16.6847	−1.39039
$145$	0	0
$146$	6.63068	0.548759
$147$	2.56155	0.211273
$148$	−2.63068	−0.216241
$149$	−4.24621	−0.347863	−0.173932	−	0.984758i	$-0.555647\pi$
$149$	−4.24621	−0.347863	−0.173932	+	0.984758i	$0.555647\pi$
$150$	0	0
$151$	21.9309	1.78471	0.892354	−	0.451335i	$-0.149052\pi$
$151$	21.9309	1.78471	0.892354	+	0.451335i	$0.149052\pi$
$152$	2.73863	0.222133
$153$	16.2462	1.31343
$154$	−4.00000	−0.322329
$155$	0	0
$156$	−5.12311	−0.410177
$157$	−3.75379	−0.299585	−0.149792	−	0.988717i	$-0.547861\pi$
$157$	−3.75379	−0.299585	−0.149792	+	0.988717i	$0.547861\pi$
$158$	10.2462	0.815145
$159$	−8.00000	−0.634441
$160$	0	0
$161$	5.12311	0.403757
$162$	10.9309	0.858810
$163$	−1.12311	−0.0879684	−0.0439842	−	0.999032i	$-0.514005\pi$
$163$	−1.12311	−0.0879684	−0.0439842	+	0.999032i	$0.514005\pi$
$164$	−1.36932	−0.106926
$165$	0	0
$166$	6.24621	0.484800
$167$	−21.9309	−1.69706	−0.848531	−	0.529146i	$-0.822512\pi$
$167$	−21.9309	−1.69706	−0.848531	+	0.529146i	$0.822512\pi$
$168$	6.24621	0.481906
$169$	7.80776	0.600597
$170$	0	0
$171$	4.00000	0.305888
$172$	−4.00000	−0.304997
$173$	8.56155	0.650923	0.325461	−	0.945555i	$-0.394480\pi$
$173$	8.56155	0.650923	0.325461	+	0.945555i	$0.394480\pi$
$174$	22.7386	1.72381
$175$	0	0
$176$	−12.0000	−0.904534
$177$	−10.2462	−0.770152
$178$	−11.1231	−0.833712
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	0	0
$181$	23.6155	1.75533	0.877664	−	0.479276i	$-0.159101\pi$
$181$	23.6155	1.75533	0.877664	+	0.479276i	$0.159101\pi$
$182$	7.12311	0.528000
$183$	−24.0000	−1.77413
$184$	12.4924	0.920954
$185$	0	0
$186$	0	0
$187$	11.6847	0.854467
$188$	−1.61553	−0.117824
$189$	1.43845	0.104632
$190$	0	0
$191$	−9.43845	−0.682942	−0.341471	−	0.939892i	$-0.610925\pi$
$191$	−9.43845	−0.682942	−0.341471	+	0.939892i	$0.610925\pi$
$192$	14.2462	1.02813
$193$	5.36932	0.386492	0.193246	−	0.981150i	$-0.438098\pi$
$193$	5.36932	0.386492	0.193246	+	0.981150i	$0.438098\pi$
$194$	−23.1231	−1.66014
$195$	0	0
$196$	0.438447	0.0313177
$197$	7.12311	0.507500	0.253750	−	0.967270i	$-0.418336\pi$
$197$	7.12311	0.507500	0.253750	+	0.967270i	$0.418336\pi$
$198$	−14.2462	−1.01243
$199$	−18.2462	−1.29344	−0.646720	−	0.762728i	$-0.723860\pi$
$199$	−18.2462	−1.29344	−0.646720	+	0.762728i	$0.723860\pi$
$200$	0	0
$201$	16.0000	1.12855
$202$	−0.384472	−0.0270513
$203$	−5.68466	−0.398985
$204$	5.12311	0.358689
$205$	0	0
$206$	2.24621	0.156501
$207$	18.2462	1.26820
$208$	21.3693	1.48170
$209$	2.87689	0.198999
$210$	0	0
$211$	−23.0540	−1.58710	−0.793551	−	0.608504i	$-0.791770\pi$
$211$	−23.0540	−1.58710	−0.793551	+	0.608504i	$0.791770\pi$
$212$	−1.36932	−0.0940451
$213$	20.4924	1.40412
$214$	−17.7538	−1.21362
$215$	0	0
$216$	3.50758	0.238660
$217$	0	0
$218$	−27.6155	−1.87036
$219$	−10.8769	−0.734992
$220$	0	0
$221$	−20.8078	−1.39968
$222$	24.0000	1.61077
$223$	6.56155	0.439394	0.219697	−	0.975568i	$-0.429493\pi$
$223$	6.56155	0.439394	0.219697	+	0.975568i	$0.429493\pi$
$224$	2.43845	0.162926
$225$	0	0
$226$	−21.8617	−1.45422
$227$	−23.6847	−1.57201	−0.786003	−	0.618223i	$-0.787853\pi$
$227$	−23.6847	−1.57201	−0.786003	+	0.618223i	$0.787853\pi$
$228$	1.26137	0.0835360
$229$	19.1231	1.26369	0.631845	−	0.775095i	$-0.282298\pi$
$229$	19.1231	1.26369	0.631845	+	0.775095i	$0.282298\pi$
$230$	0	0
$231$	6.56155	0.431718
$232$	−13.8617	−0.910068
$233$	3.12311	0.204601	0.102301	−	0.994754i	$-0.467380\pi$
$233$	3.12311	0.204601	0.102301	+	0.994754i	$0.467380\pi$
$234$	25.3693	1.65844
$235$	0	0
$236$	−1.75379	−0.114162
$237$	−16.8078	−1.09178
$238$	−7.12311	−0.461722
$239$	−0.807764	−0.0522499	−0.0261250	−	0.999659i	$-0.508317\pi$
$239$	−0.807764	−0.0522499	−0.0261250	+	0.999659i	$0.508317\pi$
$240$	0	0
$241$	12.2462	0.788848	0.394424	−	0.918929i	$-0.370944\pi$
$241$	12.2462	0.788848	0.394424	+	0.918929i	$0.370944\pi$
$242$	6.93087	0.445533
$243$	−22.2462	−1.42710
$244$	−4.10795	−0.262985
$245$	0	0
$246$	12.4924	0.796488
$247$	−5.12311	−0.325975
$248$	0	0
$249$	−10.2462	−0.649327
$250$	0	0
$251$	−17.1231	−1.08080	−0.540400	−	0.841408i	$-0.681727\pi$
$251$	−17.1231	−1.08080	−0.540400	+	0.841408i	$0.681727\pi$
$252$	1.56155	0.0983686
$253$	13.1231	0.825043
$254$	16.0000	1.00393
$255$	0	0
$256$	10.0540	0.628373
$257$	−22.4924	−1.40304	−0.701519	−	0.712650i	$-0.747495\pi$
$257$	−22.4924	−1.40304	−0.701519	+	0.712650i	$0.747495\pi$
$258$	36.4924	2.27192
$259$	−6.00000	−0.372822
$260$	0	0
$261$	−20.2462	−1.25321
$262$	14.2462	0.880134
$263$	21.1231	1.30251	0.651253	−	0.758860i	$-0.274244\pi$
$263$	21.1231	1.30251	0.651253	+	0.758860i	$0.274244\pi$
$264$	16.0000	0.984732
$265$	0	0
$266$	−1.75379	−0.107532
$267$	18.2462	1.11665
$268$	2.73863	0.167289
$269$	28.7386	1.75223	0.876113	−	0.482106i	$-0.160128\pi$
$269$	28.7386	1.75223	0.876113	+	0.482106i	$0.160128\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	−21.3693	−1.29571
$273$	−11.6847	−0.707188
$274$	−13.8617	−0.837418
$275$	0	0
$276$	5.75379	0.346337
$277$	−16.2462	−0.976140	−0.488070	−	0.872804i	$-0.662299\pi$
$277$	−16.2462	−0.976140	−0.488070	+	0.872804i	$0.662299\pi$
$278$	10.7386	0.644060
$279$	0	0
$280$	0	0
$281$	16.5616	0.987979	0.493990	−	0.869468i	$-0.335538\pi$
$281$	16.5616	0.987979	0.493990	+	0.869468i	$0.335538\pi$
$282$	14.7386	0.877673
$283$	23.6847	1.40791	0.703953	−	0.710246i	$-0.251416\pi$
$283$	23.6847	1.40791	0.703953	+	0.710246i	$0.251416\pi$
$284$	3.50758	0.208136
$285$	0	0
$286$	18.2462	1.07892
$287$	−3.12311	−0.184351
$288$	8.68466	0.511748
$289$	3.80776	0.223986
$290$	0	0
$291$	37.9309	2.22355
$292$	−1.86174	−0.108950
$293$	−9.68466	−0.565784	−0.282892	−	0.959152i	$-0.591294\pi$
$293$	−9.68466	−0.565784	−0.282892	+	0.959152i	$0.591294\pi$
$294$	−4.00000	−0.233285
$295$	0	0
$296$	−14.6307	−0.850391
$297$	3.68466	0.213806
$298$	6.63068	0.384105
$299$	−23.3693	−1.35148
$300$	0	0
$301$	−9.12311	−0.525847
$302$	−34.2462	−1.97065
$303$	0.630683	0.0362318
$304$	−5.26137	−0.301760
$305$	0	0
$306$	−25.3693	−1.45027
$307$	31.6847	1.80834	0.904169	−	0.427174i	$-0.140491\pi$
$307$	31.6847	1.80834	0.904169	+	0.427174i	$0.140491\pi$
$308$	1.12311	0.0639949
$309$	−3.68466	−0.209613
$310$	0	0
$311$	−9.61553	−0.545247	−0.272623	−	0.962121i	$-0.587891\pi$
$311$	−9.61553	−0.545247	−0.272623	+	0.962121i	$0.587891\pi$
$312$	−28.4924	−1.61307
$313$	−31.3002	−1.76919	−0.884596	−	0.466359i	$-0.845566\pi$
$313$	−31.3002	−1.76919	−0.884596	+	0.466359i	$0.845566\pi$
$314$	5.86174	0.330797
$315$	0	0
$316$	−2.87689	−0.161838
$317$	22.4924	1.26330	0.631650	−	0.775254i	$-0.282378\pi$
$317$	22.4924	1.26330	0.631650	+	0.775254i	$0.282378\pi$
$318$	12.4924	0.700540
$319$	−14.5616	−0.815290
$320$	0	0
$321$	29.1231	1.62549
$322$	−8.00000	−0.445823
$323$	5.12311	0.285057
$324$	−3.06913	−0.170507
$325$	0	0
$326$	1.75379	0.0971334
$327$	45.3002	2.50511
$328$	−7.61553	−0.420497
$329$	−3.68466	−0.203142
$330$	0	0
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	−1.75379	−0.0962517
$333$	−21.3693	−1.17103
$334$	34.2462	1.87387
$335$	0	0
$336$	−12.0000	−0.654654
$337$	34.4924	1.87892	0.939461	−	0.342656i	$-0.111326\pi$
$337$	34.4924	1.87892	0.939461	+	0.342656i	$0.111326\pi$
$338$	−12.1922	−0.663170
$339$	35.8617	1.94774
$340$	0	0
$341$	0	0
$342$	−6.24621	−0.337756
$343$	1.00000	0.0539949
$344$	−22.2462	−1.19944
$345$	0	0
$346$	−13.3693	−0.718739
$347$	1.12311	0.0602915	0.0301457	−	0.999546i	$-0.490403\pi$
$347$	1.12311	0.0602915	0.0301457	+	0.999546i	$0.490403\pi$
$348$	−6.38447	−0.342244
$349$	−22.4924	−1.20399	−0.601996	−	0.798499i	$-0.705628\pi$
$349$	−22.4924	−1.20399	−0.601996	+	0.798499i	$0.705628\pi$
$350$	0	0
$351$	−6.56155	−0.350230
$352$	6.24621	0.332924
$353$	14.8078	0.788138	0.394069	−	0.919081i	$-0.371067\pi$
$353$	14.8078	0.788138	0.394069	+	0.919081i	$0.371067\pi$
$354$	16.0000	0.850390
$355$	0	0
$356$	3.12311	0.165524
$357$	11.6847	0.618418
$358$	−31.2311	−1.65061
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	0	0
$361$	−17.7386	−0.933612
$362$	−36.8769	−1.93821
$363$	−11.3693	−0.596734
$364$	−2.00000	−0.104828
$365$	0	0
$366$	37.4773	1.95897
$367$	−3.68466	−0.192338	−0.0961688	−	0.995365i	$-0.530659\pi$
$367$	−3.68466	−0.192338	−0.0961688	+	0.995365i	$0.530659\pi$
$368$	−24.0000	−1.25109
$369$	−11.1231	−0.579046
$370$	0	0
$371$	−3.12311	−0.162144
$372$	0	0
$373$	−29.3693	−1.52069	−0.760343	−	0.649522i	$-0.774969\pi$
$373$	−29.3693	−1.52069	−0.760343	+	0.649522i	$0.774969\pi$
$374$	−18.2462	−0.943489
$375$	0	0
$376$	−8.98485	−0.463358
$377$	25.9309	1.33551
$378$	−2.24621	−0.115533
$379$	16.4924	0.847159	0.423579	−	0.905859i	$-0.360773\pi$
$379$	16.4924	0.847159	0.423579	+	0.905859i	$0.360773\pi$
$380$	0	0
$381$	−26.2462	−1.34463
$382$	14.7386	0.754094
$383$	10.2462	0.523557	0.261778	−	0.965128i	$-0.415691\pi$
$383$	10.2462	0.523557	0.261778	+	0.965128i	$0.415691\pi$
$384$	−34.7386	−1.77275
$385$	0	0
$386$	−8.38447	−0.426758
$387$	−32.4924	−1.65168
$388$	6.49242	0.329603
$389$	3.93087	0.199303	0.0996515	−	0.995022i	$-0.468227\pi$
$389$	3.93087	0.199303	0.0996515	+	0.995022i	$0.468227\pi$
$390$	0	0
$391$	23.3693	1.18184
$392$	2.43845	0.123160
$393$	−23.3693	−1.17883
$394$	−11.1231	−0.560374
$395$	0	0
$396$	4.00000	0.201008
$397$	−23.4384	−1.17634	−0.588171	−	0.808737i	$-0.700152\pi$
$397$	−23.4384	−1.17634	−0.588171	+	0.808737i	$0.700152\pi$
$398$	28.4924	1.42820
$399$	2.87689	0.144025
$400$	0	0
$401$	27.4384	1.37021	0.685105	−	0.728444i	$-0.259756\pi$
$401$	27.4384	1.37021	0.685105	+	0.728444i	$0.259756\pi$
$402$	−24.9848	−1.24613
$403$	0	0
$404$	0.107951	0.00537074
$405$	0	0
$406$	8.87689	0.440553
$407$	−15.3693	−0.761829
$408$	28.4924	1.41059
$409$	−26.4924	−1.30997	−0.654983	−	0.755644i	$-0.727324\pi$
$409$	−26.4924	−1.30997	−0.654983	+	0.755644i	$0.727324\pi$
$410$	0	0
$411$	22.7386	1.12161
$412$	−0.630683	−0.0310715
$413$	−4.00000	−0.196827
$414$	−28.4924	−1.40033
$415$	0	0
$416$	−11.1231	−0.545355
$417$	−17.6155	−0.862636
$418$	−4.49242	−0.219732
$419$	9.75379	0.476504	0.238252	−	0.971203i	$-0.423426\pi$
$419$	9.75379	0.476504	0.238252	+	0.971203i	$0.423426\pi$
$420$	0	0
$421$	9.68466	0.472001	0.236001	−	0.971753i	$-0.424163\pi$
$421$	9.68466	0.472001	0.236001	+	0.971753i	$0.424163\pi$
$422$	36.0000	1.75245
$423$	−13.1231	−0.638067
$424$	−7.61553	−0.369843
$425$	0	0
$426$	−32.0000	−1.55041
$427$	−9.36932	−0.453413
$428$	4.98485	0.240952
$429$	−29.9309	−1.44508
$430$	0	0
$431$	0.807764	0.0389086	0.0194543	−	0.999811i	$-0.493807\pi$
$431$	0.807764	0.0389086	0.0194543	+	0.999811i	$0.493807\pi$
$432$	−6.73863	−0.324213
$433$	8.24621	0.396288	0.198144	−	0.980173i	$-0.436509\pi$
$433$	8.24621	0.396288	0.198144	+	0.980173i	$0.436509\pi$
$434$	0	0
$435$	0	0
$436$	7.75379	0.371339
$437$	5.75379	0.275241
$438$	16.9848	0.811567
$439$	−15.3693	−0.733537	−0.366769	−	0.930312i	$-0.619536\pi$
$439$	−15.3693	−0.733537	−0.366769	+	0.930312i	$0.619536\pi$
$440$	0	0
$441$	3.56155	0.169598
$442$	32.4924	1.54551
$443$	27.3693	1.30036	0.650178	−	0.759782i	$-0.274694\pi$
$443$	27.3693	1.30036	0.650178	+	0.759782i	$0.274694\pi$
$444$	−6.73863	−0.319801
$445$	0	0
$446$	−10.2462	−0.485172
$447$	−10.8769	−0.514459
$448$	5.56155	0.262759
$449$	18.8078	0.887593	0.443797	−	0.896128i	$-0.353631\pi$
$449$	18.8078	0.887593	0.443797	+	0.896128i	$0.353631\pi$
$450$	0	0
$451$	−8.00000	−0.376705
$452$	6.13826	0.288719
$453$	56.1771	2.63943
$454$	36.9848	1.73578
$455$	0	0
$456$	7.01515	0.328515
$457$	8.87689	0.415244	0.207622	−	0.978209i	$-0.433428\pi$
$457$	8.87689	0.415244	0.207622	+	0.978209i	$0.433428\pi$
$458$	−29.8617	−1.39535
$459$	6.56155	0.306267
$460$	0	0
$461$	−4.87689	−0.227140	−0.113570	−	0.993530i	$-0.536229\pi$
$461$	−4.87689	−0.227140	−0.113570	+	0.993530i	$0.536229\pi$
$462$	−10.2462	−0.476697
$463$	20.4924	0.952364	0.476182	−	0.879347i	$-0.342020\pi$
$463$	20.4924	0.952364	0.476182	+	0.879347i	$0.342020\pi$
$464$	26.6307	1.23630
$465$	0	0
$466$	−4.87689	−0.225918
$467$	−26.5616	−1.22912	−0.614561	−	0.788869i	$-0.710667\pi$
$467$	−26.5616	−1.22912	−0.614561	+	0.788869i	$0.710667\pi$
$468$	−7.12311	−0.329266
$469$	6.24621	0.288423
$470$	0	0
$471$	−9.61553	−0.443060
$472$	−9.75379	−0.448955
$473$	−23.3693	−1.07452
$474$	26.2462	1.20553
$475$	0	0
$476$	2.00000	0.0916698
$477$	−11.1231	−0.509292
$478$	1.26137	0.0576935
$479$	13.1231	0.599610	0.299805	−	0.954001i	$-0.403078\pi$
$479$	13.1231	0.599610	0.299805	+	0.954001i	$0.403078\pi$
$480$	0	0
$481$	27.3693	1.24793
$482$	−19.1231	−0.871034
$483$	13.1231	0.597122
$484$	−1.94602	−0.0884557
$485$	0	0
$486$	34.7386	1.57578
$487$	−5.12311	−0.232150	−0.116075	−	0.993240i	$-0.537031\pi$
$487$	−5.12311	−0.232150	−0.116075	+	0.993240i	$0.537031\pi$
$488$	−22.8466	−1.03422
$489$	−2.87689	−0.130098
$490$	0	0
$491$	4.17708	0.188509	0.0942545	−	0.995548i	$-0.469953\pi$
$491$	4.17708	0.188509	0.0942545	+	0.995548i	$0.469953\pi$
$492$	−3.50758	−0.158134
$493$	−25.9309	−1.16787
$494$	8.00000	0.359937
$495$	0	0
$496$	0	0
$497$	8.00000	0.358849
$498$	16.0000	0.716977
$499$	−4.17708	−0.186992	−0.0934959	−	0.995620i	$-0.529804\pi$
$499$	−4.17708	−0.186992	−0.0934959	+	0.995620i	$0.529804\pi$
$500$	0	0
$501$	−56.1771	−2.50981
$502$	26.7386	1.19340
$503$	−10.0691	−0.448960	−0.224480	−	0.974479i	$-0.572068\pi$
$503$	−10.0691	−0.448960	−0.224480	+	0.974479i	$0.572068\pi$
$504$	8.68466	0.386845
$505$	0	0
$506$	−20.4924	−0.910999
$507$	20.0000	0.888231
$508$	−4.49242	−0.199319
$509$	−28.2462	−1.25199	−0.625996	−	0.779827i	$-0.715307\pi$
$509$	−28.2462	−1.25199	−0.625996	+	0.779827i	$0.715307\pi$
$510$	0	0
$511$	−4.24621	−0.187841
$512$	11.4233	0.504843
$513$	1.61553	0.0713273
$514$	35.1231	1.54921
$515$	0	0
$516$	−10.2462	−0.451064
$517$	−9.43845	−0.415102
$518$	9.36932	0.411664
$519$	21.9309	0.962658
$520$	0	0
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	31.6155	1.38377
$523$	−7.50758	−0.328283	−0.164142	−	0.986437i	$-0.552485\pi$
$523$	−7.50758	−0.328283	−0.164142	+	0.986437i	$0.552485\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	−32.9848	−1.43821
$527$	0	0
$528$	−30.7386	−1.33773
$529$	3.24621	0.141140
$530$	0	0
$531$	−14.2462	−0.618233
$532$	0.492423	0.0213492
$533$	14.2462	0.617072
$534$	−28.4924	−1.23299
$535$	0	0
$536$	15.2311	0.657881
$537$	51.2311	2.21078
$538$	−44.8769	−1.93478
$539$	2.56155	0.110334
$540$	0	0
$541$	−17.1922	−0.739152	−0.369576	−	0.929201i	$-0.620497\pi$
$541$	−17.1922	−0.739152	−0.369576	+	0.929201i	$0.620497\pi$
$542$	24.9848	1.07319
$543$	60.4924	2.59598
$544$	11.1231	0.476899
$545$	0	0
$546$	18.2462	0.780866
$547$	−14.2462	−0.609124	−0.304562	−	0.952493i	$-0.598510\pi$
$547$	−14.2462	−0.609124	−0.304562	+	0.952493i	$0.598510\pi$
$548$	3.89205	0.166260
$549$	−33.3693	−1.42417
$550$	0	0
$551$	−6.38447	−0.271988
$552$	32.0000	1.36201
$553$	−6.56155	−0.279026
$554$	25.3693	1.07784
$555$	0	0
$556$	−3.01515	−0.127871
$557$	4.87689	0.206641	0.103320	−	0.994648i	$-0.467053\pi$
$557$	4.87689	0.206641	0.103320	+	0.994648i	$0.467053\pi$
$558$	0	0
$559$	41.6155	1.76015
$560$	0	0
$561$	29.9309	1.26368
$562$	−25.8617	−1.09091
$563$	28.0000	1.18006	0.590030	−	0.807382i	$-0.299116\pi$
$563$	28.0000	1.18006	0.590030	+	0.807382i	$0.299116\pi$
$564$	−4.13826	−0.174252
$565$	0	0
$566$	−36.9848	−1.55459
$567$	−7.00000	−0.293972
$568$	19.5076	0.818520
$569$	34.9848	1.46664	0.733320	−	0.679883i	$-0.237969\pi$
$569$	34.9848	1.46664	0.733320	+	0.679883i	$0.237969\pi$
$570$	0	0
$571$	7.50758	0.314182	0.157091	−	0.987584i	$-0.449788\pi$
$571$	7.50758	0.314182	0.157091	+	0.987584i	$0.449788\pi$
$572$	−5.12311	−0.214208
$573$	−24.1771	−1.01001
$574$	4.87689	0.203558
$575$	0	0
$576$	19.8078	0.825324
$577$	−13.0540	−0.543444	−0.271722	−	0.962376i	$-0.587593\pi$
$577$	−13.0540	−0.543444	−0.271722	+	0.962376i	$0.587593\pi$
$578$	−5.94602	−0.247322
$579$	13.7538	0.571588
$580$	0	0
$581$	−4.00000	−0.165948
$582$	−59.2311	−2.45521
$583$	−8.00000	−0.331326
$584$	−10.3542	−0.428458
$585$	0	0
$586$	15.1231	0.624730
$587$	9.75379	0.402582	0.201291	−	0.979531i	$-0.435486\pi$
$587$	9.75379	0.402582	0.201291	+	0.979531i	$0.435486\pi$
$588$	1.12311	0.0463161
$589$	0	0
$590$	0	0
$591$	18.2462	0.750549
$592$	28.1080	1.15523
$593$	23.4384	0.962502	0.481251	−	0.876583i	$-0.340183\pi$
$593$	23.4384	0.962502	0.481251	+	0.876583i	$0.340183\pi$
$594$	−5.75379	−0.236081
$595$	0	0
$596$	−1.86174	−0.0762598
$597$	−46.7386	−1.91288
$598$	36.4924	1.49229
$599$	8.80776	0.359875	0.179938	−	0.983678i	$-0.442410\pi$
$599$	8.80776	0.359875	0.179938	+	0.983678i	$0.442410\pi$
$600$	0	0
$601$	−26.4924	−1.08065	−0.540324	−	0.841457i	$-0.681698\pi$
$601$	−26.4924	−1.08065	−0.540324	+	0.841457i	$0.681698\pi$
$602$	14.2462	0.580632
$603$	22.2462	0.905936
$604$	9.61553	0.391250
$605$	0	0
$606$	−0.984845	−0.0400066
$607$	4.94602	0.200753	0.100376	−	0.994950i	$-0.467995\pi$
$607$	4.94602	0.200753	0.100376	+	0.994950i	$0.467995\pi$
$608$	2.73863	0.111066
$609$	−14.5616	−0.590064
$610$	0	0
$611$	16.8078	0.679969
$612$	7.12311	0.287934
$613$	8.73863	0.352950	0.176475	−	0.984305i	$-0.443531\pi$
$613$	8.73863	0.352950	0.176475	+	0.984305i	$0.443531\pi$
$614$	−49.4773	−1.99674
$615$	0	0
$616$	6.24621	0.251667
$617$	−15.7538	−0.634224	−0.317112	−	0.948388i	$-0.602713\pi$
$617$	−15.7538	−0.634224	−0.317112	+	0.948388i	$0.602713\pi$
$618$	5.75379	0.231451
$619$	−42.1080	−1.69246	−0.846231	−	0.532817i	$-0.821134\pi$
$619$	−42.1080	−1.69246	−0.846231	+	0.532817i	$0.821134\pi$
$620$	0	0
$621$	7.36932	0.295720
$622$	15.0152	0.602053
$623$	7.12311	0.285381
$624$	54.7386	2.19130
$625$	0	0
$626$	48.8769	1.95351
$627$	7.36932	0.294302
$628$	−1.64584	−0.0656761
$629$	−27.3693	−1.09129
$630$	0	0
$631$	8.80776	0.350632	0.175316	−	0.984512i	$-0.443905\pi$
$631$	8.80776	0.350632	0.175316	+	0.984512i	$0.443905\pi$
$632$	−16.0000	−0.636446
$633$	−59.0540	−2.34718
$634$	−35.1231	−1.39492
$635$	0	0
$636$	−3.50758	−0.139084
$637$	−4.56155	−0.180735
$638$	22.7386	0.900231
$639$	28.4924	1.12714
$640$	0	0
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	−45.4773	−1.79484
$643$	2.56155	0.101018	0.0505089	−	0.998724i	$-0.483916\pi$
$643$	2.56155	0.101018	0.0505089	+	0.998724i	$0.483916\pi$
$644$	2.24621	0.0885131
$645$	0	0
$646$	−8.00000	−0.314756
$647$	−3.50758	−0.137897	−0.0689486	−	0.997620i	$-0.521964\pi$
$647$	−3.50758	−0.137897	−0.0689486	+	0.997620i	$0.521964\pi$
$648$	−17.0691	−0.670539
$649$	−10.2462	−0.402199
$650$	0	0
$651$	0	0
$652$	−0.492423	−0.0192848
$653$	−49.2311	−1.92656	−0.963280	−	0.268499i	$-0.913473\pi$
$653$	−49.2311	−1.92656	−0.963280	+	0.268499i	$0.913473\pi$
$654$	−70.7386	−2.76610
$655$	0	0
$656$	14.6307	0.571232
$657$	−15.1231	−0.590009
$658$	5.75379	0.224306
$659$	−36.1771	−1.40926	−0.704629	−	0.709575i	$-0.748887\pi$
$659$	−36.1771	−1.40926	−0.704629	+	0.709575i	$0.748887\pi$
$660$	0	0
$661$	3.12311	0.121475	0.0607374	−	0.998154i	$-0.480655\pi$
$661$	3.12311	0.121475	0.0607374	+	0.998154i	$0.480655\pi$
$662$	−18.7386	−0.728298
$663$	−53.3002	−2.07001
$664$	−9.75379	−0.378520
$665$	0	0
$666$	33.3693	1.29303
$667$	−29.1231	−1.12765
$668$	−9.61553	−0.372036
$669$	16.8078	0.649826
$670$	0	0
$671$	−24.0000	−0.926510
$672$	6.24621	0.240953
$673$	25.8617	0.996897	0.498448	−	0.866919i	$-0.333903\pi$
$673$	25.8617	0.996897	0.498448	+	0.866919i	$0.333903\pi$
$674$	−53.8617	−2.07468
$675$	0	0
$676$	3.42329	0.131665
$677$	23.9309	0.919738	0.459869	−	0.887987i	$-0.347896\pi$
$677$	23.9309	0.919738	0.459869	+	0.887987i	$0.347896\pi$
$678$	−56.0000	−2.15067
$679$	14.8078	0.568270
$680$	0	0
$681$	−60.6695	−2.32486
$682$	0	0
$683$	−42.7386	−1.63535	−0.817674	−	0.575681i	$-0.804737\pi$
$683$	−42.7386	−1.63535	−0.817674	+	0.575681i	$0.804737\pi$
$684$	1.75379	0.0670578
$685$	0	0
$686$	−1.56155	−0.0596204
$687$	48.9848	1.86889
$688$	42.7386	1.62940
$689$	14.2462	0.542737
$690$	0	0
$691$	8.49242	0.323067	0.161533	−	0.986867i	$-0.448356\pi$
$691$	8.49242	0.323067	0.161533	+	0.986867i	$0.448356\pi$
$692$	3.75379	0.142698
$693$	9.12311	0.346558
$694$	−1.75379	−0.0665729
$695$	0	0
$696$	−35.5076	−1.34591
$697$	−14.2462	−0.539614
$698$	35.1231	1.32943
$699$	8.00000	0.302588
$700$	0	0
$701$	0.0691303	0.00261102	0.00130551	−	0.999999i	$-0.499584\pi$
$701$	0.0691303	0.00261102	0.00130551	+	0.999999i	$0.499584\pi$
$702$	10.2462	0.386718
$703$	−6.73863	−0.254152
$704$	14.2462	0.536924
$705$	0	0
$706$	−23.1231	−0.870250
$707$	0.246211	0.00925973
$708$	−4.49242	−0.168836
$709$	−18.1771	−0.682655	−0.341327	−	0.939945i	$-0.610876\pi$
$709$	−18.1771	−0.682655	−0.341327	+	0.939945i	$0.610876\pi$
$710$	0	0
$711$	−23.3693	−0.876418
$712$	17.3693	0.650943
$713$	0	0
$714$	−18.2462	−0.682847
$715$	0	0
$716$	8.76894	0.327711
$717$	−2.06913	−0.0772731
$718$	−12.4924	−0.466213
$719$	49.6155	1.85035	0.925173	−	0.379544i	$-0.123919\pi$
$719$	49.6155	1.85035	0.925173	+	0.379544i	$0.123919\pi$
$720$	0	0
$721$	−1.43845	−0.0535706
$722$	27.6998	1.03088
$723$	31.3693	1.16664
$724$	10.3542	0.384809
$725$	0	0
$726$	17.7538	0.658905
$727$	−19.5076	−0.723496	−0.361748	−	0.932276i	$-0.617820\pi$
$727$	−19.5076	−0.723496	−0.361748	+	0.932276i	$0.617820\pi$
$728$	−11.1231	−0.412250
$729$	−35.9848	−1.33277
$730$	0	0
$731$	−41.6155	−1.53921
$732$	−10.5227	−0.388931
$733$	5.68466	0.209968	0.104984	−	0.994474i	$-0.466521\pi$
$733$	5.68466	0.209968	0.104984	+	0.994474i	$0.466521\pi$
$734$	5.75379	0.212376
$735$	0	0
$736$	12.4924	0.460477
$737$	16.0000	0.589368
$738$	17.3693	0.639373
$739$	6.06913	0.223257	0.111628	−	0.993750i	$-0.464393\pi$
$739$	6.06913	0.223257	0.111628	+	0.993750i	$0.464393\pi$
$740$	0	0
$741$	−13.1231	−0.482089
$742$	4.87689	0.179036
$743$	−32.9848	−1.21010	−0.605048	−	0.796189i	$-0.706846\pi$
$743$	−32.9848	−1.21010	−0.605048	+	0.796189i	$0.706846\pi$
$744$	0	0
$745$	0	0
$746$	45.8617	1.67912
$747$	−14.2462	−0.521242
$748$	5.12311	0.187319
$749$	11.3693	0.415426
$750$	0	0
$751$	45.9309	1.67604	0.838021	−	0.545639i	$-0.183713\pi$
$751$	45.9309	1.67604	0.838021	+	0.545639i	$0.183713\pi$
$752$	17.2614	0.629457
$753$	−43.8617	−1.59841
$754$	−40.4924	−1.47465
$755$	0	0
$756$	0.630683	0.0229377
$757$	−14.6307	−0.531761	−0.265881	−	0.964006i	$-0.585663\pi$
$757$	−14.6307	−0.531761	−0.265881	+	0.964006i	$0.585663\pi$
$758$	−25.7538	−0.935420
$759$	33.6155	1.22017
$760$	0	0
$761$	31.7538	1.15107	0.575537	−	0.817776i	$-0.304793\pi$
$761$	31.7538	1.15107	0.575537	+	0.817776i	$0.304793\pi$
$762$	40.9848	1.48472
$763$	17.6847	0.640228
$764$	−4.13826	−0.149717
$765$	0	0
$766$	−16.0000	−0.578103
$767$	18.2462	0.658833
$768$	25.7538	0.929310
$769$	−9.50758	−0.342852	−0.171426	−	0.985197i	$-0.554837\pi$
$769$	−9.50758	−0.342852	−0.171426	+	0.985197i	$0.554837\pi$
$770$	0	0
$771$	−57.6155	−2.07497
$772$	2.35416	0.0847281
$773$	−8.06913	−0.290226	−0.145113	−	0.989415i	$-0.546355\pi$
$773$	−8.06913	−0.290226	−0.145113	+	0.989415i	$0.546355\pi$
$774$	50.7386	1.82376
$775$	0	0
$776$	36.1080	1.29620
$777$	−15.3693	−0.551371
$778$	−6.13826	−0.220067
$779$	−3.50758	−0.125672
$780$	0	0
$781$	20.4924	0.733277
$782$	−36.4924	−1.30497
$783$	−8.17708	−0.292225
$784$	−4.68466	−0.167309
$785$	0	0
$786$	36.4924	1.30164
$787$	3.82292	0.136272	0.0681362	−	0.997676i	$-0.478295\pi$
$787$	3.82292	0.136272	0.0681362	+	0.997676i	$0.478295\pi$
$788$	3.12311	0.111256
$789$	54.1080	1.92629
$790$	0	0
$791$	14.0000	0.497783
$792$	22.2462	0.790485
$793$	42.7386	1.51769
$794$	36.6004	1.29890
$795$	0	0
$796$	−8.00000	−0.283552
$797$	13.0540	0.462396	0.231198	−	0.972907i	$-0.425736\pi$
$797$	13.0540	0.462396	0.231198	+	0.972907i	$0.425736\pi$
$798$	−4.49242	−0.159030
$799$	−16.8078	−0.594616
$800$	0	0
$801$	25.3693	0.896381
$802$	−42.8466	−1.51297
$803$	−10.8769	−0.383837
$804$	7.01515	0.247405
$805$	0	0
$806$	0	0
$807$	73.6155	2.59139
$808$	0.600373	0.0211211
$809$	−53.5464	−1.88259	−0.941296	−	0.337584i	$-0.890390\pi$
$809$	−53.5464	−1.88259	−0.941296	+	0.337584i	$0.890390\pi$
$810$	0	0
$811$	−21.6155	−0.759024	−0.379512	−	0.925187i	$-0.623908\pi$
$811$	−21.6155	−0.759024	−0.379512	+	0.925187i	$0.623908\pi$
$812$	−2.49242	−0.0874669
$813$	−40.9848	−1.43740
$814$	24.0000	0.841200
$815$	0	0
$816$	−54.7386	−1.91624
$817$	−10.2462	−0.358470
$818$	41.3693	1.44644
$819$	−16.2462	−0.567689
$820$	0	0
$821$	40.4233	1.41078	0.705391	−	0.708818i	$-0.250771\pi$
$821$	40.4233	1.41078	0.705391	+	0.708818i	$0.250771\pi$
$822$	−35.5076	−1.23847
$823$	3.50758	0.122266	0.0611332	−	0.998130i	$-0.480529\pi$
$823$	3.50758	0.122266	0.0611332	+	0.998130i	$0.480529\pi$
$824$	−3.50758	−0.122192
$825$	0	0
$826$	6.24621	0.217333
$827$	−19.3693	−0.673537	−0.336769	−	0.941587i	$-0.609334\pi$
$827$	−19.3693	−0.673537	−0.336769	+	0.941587i	$0.609334\pi$
$828$	8.00000	0.278019
$829$	43.1231	1.49773	0.748864	−	0.662724i	$-0.230600\pi$
$829$	43.1231	1.49773	0.748864	+	0.662724i	$0.230600\pi$
$830$	0	0
$831$	−41.6155	−1.44363
$832$	−25.3693	−0.879523
$833$	4.56155	0.158048
$834$	27.5076	0.952510
$835$	0	0
$836$	1.26137	0.0436253
$837$	0	0
$838$	−15.2311	−0.526148
$839$	−37.1231	−1.28163	−0.640816	−	0.767695i	$-0.721404\pi$
$839$	−37.1231	−1.28163	−0.640816	+	0.767695i	$0.721404\pi$
$840$	0	0
$841$	3.31534	0.114322
$842$	−15.1231	−0.521177
$843$	42.4233	1.46114
$844$	−10.1080	−0.347930
$845$	0	0
$846$	20.4924	0.704544
$847$	−4.43845	−0.152507
$848$	14.6307	0.502420
$849$	60.6695	2.08217
$850$	0	0
$851$	−30.7386	−1.05371
$852$	8.98485	0.307816
$853$	56.7386	1.94269	0.971347	−	0.237666i	$-0.0763824\pi$
$853$	56.7386	1.94269	0.971347	+	0.237666i	$0.0763824\pi$
$854$	14.6307	0.500652
$855$	0	0
$856$	27.7235	0.947569
$857$	32.2462	1.10151	0.550755	−	0.834667i	$-0.314340\pi$
$857$	32.2462	1.10151	0.550755	+	0.834667i	$0.314340\pi$
$858$	46.7386	1.59563
$859$	16.4924	0.562714	0.281357	−	0.959603i	$-0.409215\pi$
$859$	16.4924	0.562714	0.281357	+	0.959603i	$0.409215\pi$
$860$	0	0
$861$	−8.00000	−0.272639
$862$	−1.26137	−0.0429623
$863$	42.2462	1.43808	0.719039	−	0.694970i	$-0.244582\pi$
$863$	42.2462	1.43808	0.719039	+	0.694970i	$0.244582\pi$
$864$	3.50758	0.119330
$865$	0	0
$866$	−12.8769	−0.437575
$867$	9.75379	0.331256
$868$	0	0
$869$	−16.8078	−0.570164
$870$	0	0
$871$	−28.4924	−0.965429
$872$	43.1231	1.46033
$873$	52.7386	1.78493
$874$	−8.98485	−0.303917
$875$	0	0
$876$	−4.76894	−0.161128
$877$	23.7538	0.802108	0.401054	−	0.916054i	$-0.368644\pi$
$877$	23.7538	0.802108	0.401054	+	0.916054i	$0.368644\pi$
$878$	24.0000	0.809961
$879$	−24.8078	−0.836745
$880$	0	0
$881$	45.8617	1.54512	0.772561	−	0.634941i	$-0.218976\pi$
$881$	45.8617	1.54512	0.772561	+	0.634941i	$0.218976\pi$
$882$	−5.56155	−0.187267
$883$	−24.4924	−0.824236	−0.412118	−	0.911131i	$-0.635211\pi$
$883$	−24.4924	−0.824236	−0.412118	+	0.911131i	$0.635211\pi$
$884$	−9.12311	−0.306843
$885$	0	0
$886$	−42.7386	−1.43583
$887$	12.4924	0.419454	0.209727	−	0.977760i	$-0.432742\pi$
$887$	12.4924	0.419454	0.209727	+	0.977760i	$0.432742\pi$
$888$	−37.4773	−1.25765
$889$	−10.2462	−0.343647
$890$	0	0
$891$	−17.9309	−0.600707
$892$	2.87689	0.0963255
$893$	−4.13826	−0.138482
$894$	16.9848	0.568058
$895$	0	0
$896$	−13.5616	−0.453060
$897$	−59.8617	−1.99873
$898$	−29.3693	−0.980067
$899$	0	0
$900$	0	0
$901$	−14.2462	−0.474610
$902$	12.4924	0.415952
$903$	−23.3693	−0.777682
$904$	34.1383	1.13542
$905$	0	0
$906$	−87.7235	−2.91442
$907$	−50.1080	−1.66381	−0.831904	−	0.554920i	$-0.812749\pi$
$907$	−50.1080	−1.66381	−0.831904	+	0.554920i	$0.812749\pi$
$908$	−10.3845	−0.344621
$909$	0.876894	0.0290848
$910$	0	0
$911$	4.49242	0.148841	0.0744203	−	0.997227i	$-0.476289\pi$
$911$	4.49242	0.148841	0.0744203	+	0.997227i	$0.476289\pi$
$912$	−13.4773	−0.446277
$913$	−10.2462	−0.339100
$914$	−13.8617	−0.458506
$915$	0	0
$916$	8.38447	0.277031
$917$	−9.12311	−0.301271
$918$	−10.2462	−0.338175
$919$	−13.3002	−0.438733	−0.219366	−	0.975643i	$-0.570399\pi$
$919$	−13.3002	−0.438733	−0.219366	+	0.975643i	$0.570399\pi$
$920$	0	0
$921$	81.1619	2.67438
$922$	7.61553	0.250804
$923$	−36.4924	−1.20116
$924$	2.87689	0.0946429
$925$	0	0
$926$	−32.0000	−1.05159
$927$	−5.12311	−0.168265
$928$	−13.8617	−0.455034
$929$	−52.1080	−1.70961	−0.854803	−	0.518952i	$-0.826322\pi$
$929$	−52.1080	−1.70961	−0.854803	+	0.518952i	$0.826322\pi$
$930$	0	0
$931$	1.12311	0.0368083
$932$	1.36932	0.0448535
$933$	−24.6307	−0.806372
$934$	41.4773	1.35718
$935$	0	0
$936$	−39.6155	−1.29487
$937$	−22.6695	−0.740580	−0.370290	−	0.928916i	$-0.620742\pi$
$937$	−22.6695	−0.740580	−0.370290	+	0.928916i	$0.620742\pi$
$938$	−9.75379	−0.318472
$939$	−80.1771	−2.61648
$940$	0	0
$941$	−13.8617	−0.451880	−0.225940	−	0.974141i	$-0.572545\pi$
$941$	−13.8617	−0.451880	−0.225940	+	0.974141i	$0.572545\pi$
$942$	15.0152	0.489220
$943$	−16.0000	−0.521032
$944$	18.7386	0.609891
$945$	0	0
$946$	36.4924	1.18647
$947$	−4.00000	−0.129983	−0.0649913	−	0.997886i	$-0.520702\pi$
$947$	−4.00000	−0.129983	−0.0649913	+	0.997886i	$0.520702\pi$
$948$	−7.36932	−0.239344
$949$	19.3693	0.628755
$950$	0	0
$951$	57.6155	1.86831
$952$	11.1231	0.360502
$953$	24.8769	0.805842	0.402921	−	0.915235i	$-0.367995\pi$
$953$	24.8769	0.805842	0.402921	+	0.915235i	$0.367995\pi$
$954$	17.3693	0.562352
$955$	0	0
$956$	−0.354162	−0.0114544
$957$	−37.3002	−1.20574
$958$	−20.4924	−0.662080
$959$	8.87689	0.286650
$960$	0	0
$961$	−31.0000	−1.00000
$962$	−42.7386	−1.37795
$963$	40.4924	1.30485
$964$	5.36932	0.172934
$965$	0	0
$966$	−20.4924	−0.659333
$967$	26.8769	0.864303	0.432151	−	0.901801i	$-0.357755\pi$
$967$	26.8769	0.864303	0.432151	+	0.901801i	$0.357755\pi$
$968$	−10.8229	−0.347862
$969$	13.1231	0.421575
$970$	0	0
$971$	−49.4773	−1.58780	−0.793901	−	0.608048i	$-0.791953\pi$
$971$	−49.4773	−1.58780	−0.793901	+	0.608048i	$0.791953\pi$
$972$	−9.75379	−0.312853
$973$	−6.87689	−0.220463
$974$	8.00000	0.256337
$975$	0	0
$976$	43.8920	1.40495
$977$	49.2311	1.57504	0.787521	−	0.616288i	$-0.211364\pi$
$977$	49.2311	1.57504	0.787521	+	0.616288i	$0.211364\pi$
$978$	4.49242	0.143652
$979$	18.2462	0.583151
$980$	0	0
$981$	62.9848	2.01095
$982$	−6.52273	−0.208149
$983$	10.4233	0.332451	0.166226	−	0.986088i	$-0.446842\pi$
$983$	10.4233	0.332451	0.166226	+	0.986088i	$0.446842\pi$
$984$	−19.5076	−0.621879
$985$	0	0
$986$	40.4924	1.28954
$987$	−9.43845	−0.300429
$988$	−2.24621	−0.0714615
$989$	−46.7386	−1.48620
$990$	0	0
$991$	−20.4924	−0.650963	−0.325482	−	0.945548i	$-0.605526\pi$
$991$	−20.4924	−0.650963	−0.325482	+	0.945548i	$0.605526\pi$
$992$	0	0
$993$	30.7386	0.975461
$994$	−12.4924	−0.396236
$995$	0	0
$996$	−4.49242	−0.142348
$997$	−9.68466	−0.306716	−0.153358	−	0.988171i	$-0.549009\pi$
$997$	−9.68466	−0.306716	−0.153358	+	0.988171i	$0.549009\pi$
$998$	6.52273	0.206473
$999$	−8.63068	−0.273063

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	175.2.a.f.1.1		2
3.2	odd	2		1575.2.a.p.1.2		2
4.3	odd	2		2800.2.a.bi.1.1		2
5.2	odd	4		175.2.b.b.99.2		4
5.3	odd	4		175.2.b.b.99.3		4
5.4	even	2		35.2.a.b.1.2	✓	2
7.6	odd	2		1225.2.a.s.1.1		2
15.2	even	4		1575.2.d.e.1324.3		4
15.8	even	4		1575.2.d.e.1324.2		4
15.14	odd	2		315.2.a.e.1.1		2
20.3	even	4		2800.2.g.t.449.1		4
20.7	even	4		2800.2.g.t.449.4		4
20.19	odd	2		560.2.a.i.1.2		2
35.4	even	6		245.2.e.i.226.1		4
35.9	even	6		245.2.e.i.116.1		4
35.13	even	4		1225.2.b.f.99.3		4
35.19	odd	6		245.2.e.h.116.1		4
35.24	odd	6		245.2.e.h.226.1		4
35.27	even	4		1225.2.b.f.99.2		4
35.34	odd	2		245.2.a.d.1.2		2
40.19	odd	2		2240.2.a.bd.1.1		2
40.29	even	2		2240.2.a.bh.1.2		2
55.54	odd	2		4235.2.a.m.1.1		2
60.59	even	2		5040.2.a.bt.1.2		2
65.64	even	2		5915.2.a.l.1.1		2
105.104	even	2		2205.2.a.x.1.1		2
140.139	even	2		3920.2.a.bs.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.2.a.b.1.2	✓	2	5.4	even	2
175.2.a.f.1.1		2	1.1	even	1	trivial
175.2.b.b.99.2		4	5.2	odd	4
175.2.b.b.99.3		4	5.3	odd	4
245.2.a.d.1.2		2	35.34	odd	2
245.2.e.h.116.1		4	35.19	odd	6
245.2.e.h.226.1		4	35.24	odd	6
245.2.e.i.116.1		4	35.9	even	6
245.2.e.i.226.1		4	35.4	even	6
315.2.a.e.1.1		2	15.14	odd	2
560.2.a.i.1.2		2	20.19	odd	2
1225.2.a.s.1.1		2	7.6	odd	2
1225.2.b.f.99.2		4	35.27	even	4
1225.2.b.f.99.3		4	35.13	even	4
1575.2.a.p.1.2		2	3.2	odd	2
1575.2.d.e.1324.2		4	15.8	even	4
1575.2.d.e.1324.3		4	15.2	even	4
2205.2.a.x.1.1		2	105.104	even	2
2240.2.a.bd.1.1		2	40.19	odd	2
2240.2.a.bh.1.2		2	40.29	even	2
2800.2.a.bi.1.1		2	4.3	odd	2
2800.2.g.t.449.1		4	20.3	even	4
2800.2.g.t.449.4		4	20.7	even	4
3920.2.a.bs.1.1		2	140.139	even	2
4235.2.a.m.1.1		2	55.54	odd	2
5040.2.a.bt.1.2		2	60.59	even	2
5915.2.a.l.1.1		2	65.64	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 175 = 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	175.a (trivial)

\(f(q)\)	\(=\)	\(q-1.56155 q^{2} +2.56155 q^{3} +0.438447 q^{4} -4.00000 q^{6} +1.00000 q^{7} +2.43845 q^{8} +3.56155 q^{9} +2.56155 q^{11} +1.12311 q^{12} -4.56155 q^{13} -1.56155 q^{14} -4.68466 q^{16} +4.56155 q^{17} -5.56155 q^{18} +1.12311 q^{19} +2.56155 q^{21} -4.00000 q^{22} +5.12311 q^{23} +6.24621 q^{24} +7.12311 q^{26} +1.43845 q^{27} +0.438447 q^{28} -5.68466 q^{29} +2.43845 q^{32} +6.56155 q^{33} -7.12311 q^{34} +1.56155 q^{36} -6.00000 q^{37} -1.75379 q^{38} -11.6847 q^{39} -3.12311 q^{41} -4.00000 q^{42} -9.12311 q^{43} +1.12311 q^{44} -8.00000 q^{46} -3.68466 q^{47} -12.0000 q^{48} +1.00000 q^{49} +11.6847 q^{51} -2.00000 q^{52} -3.12311 q^{53} -2.24621 q^{54} +2.43845 q^{56} +2.87689 q^{57} +8.87689 q^{58} -4.00000 q^{59} -9.36932 q^{61} +3.56155 q^{63} +5.56155 q^{64} -10.2462 q^{66} +6.24621 q^{67} +2.00000 q^{68} +13.1231 q^{69} +8.00000 q^{71} +8.68466 q^{72} -4.24621 q^{73} +9.36932 q^{74} +0.492423 q^{76} +2.56155 q^{77} +18.2462 q^{78} -6.56155 q^{79} -7.00000 q^{81} +4.87689 q^{82} -4.00000 q^{83} +1.12311 q^{84} +14.2462 q^{86} -14.5616 q^{87} +6.24621 q^{88} +7.12311 q^{89} -4.56155 q^{91} +2.24621 q^{92} +5.75379 q^{94} +6.24621 q^{96} +14.8078 q^{97} -1.56155 q^{98} +9.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + q^{3} + 5 q^{4} - 8 q^{6} + 2 q^{7} + 9 q^{8} + 3 q^{9} + q^{11} - 6 q^{12} - 5 q^{13} + q^{14} + 3 q^{16} + 5 q^{17} - 7 q^{18} - 6 q^{19} + q^{21} - 8 q^{22} + 2 q^{23} - 4 q^{24} + 6 q^{26}+ \cdots + 10 q^{99}+O(q^{100}) \) 2 * q + q^2 + q^3 + 5 * q^4 - 8 * q^6 + 2 * q^7 + 9 * q^8 + 3 * q^9 + q^11 - 6 * q^12 - 5 * q^13 + q^14 + 3 * q^16 + 5 * q^17 - 7 * q^18 - 6 * q^19 + q^21 - 8 * q^22 + 2 * q^23 - 4 * q^24 + 6 * q^26 + 7 * q^27 + 5 * q^28 + q^29 + 9 * q^32 + 9 * q^33 - 6 * q^34 - q^36 - 12 * q^37 - 20 * q^38 - 11 * q^39 + 2 * q^41 - 8 * q^42 - 10 * q^43 - 6 * q^44 - 16 * q^46 + 5 * q^47 - 24 * q^48 + 2 * q^49 + 11 * q^51 - 4 * q^52 + 2 * q^53 + 12 * q^54 + 9 * q^56 + 14 * q^57 + 26 * q^58 - 8 * q^59 + 6 * q^61 + 3 * q^63 + 7 * q^64 - 4 * q^66 - 4 * q^67 + 4 * q^68 + 18 * q^69 + 16 * q^71 + 5 * q^72 + 8 * q^73 - 6 * q^74 - 32 * q^76 + q^77 + 20 * q^78 - 9 * q^79 - 14 * q^81 + 18 * q^82 - 8 * q^83 - 6 * q^84 + 12 * q^86 - 25 * q^87 - 4 * q^88 + 6 * q^89 - 5 * q^91 - 12 * q^92 + 28 * q^94 - 4 * q^96 + 9 * q^97 + q^98 + 10 * q^99

Introduction

L-functions

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